I am looking for a purely mathematical example. I tried looking for a set of symmetric matrices $\{F_1,F_2\}$ such that $F_1+F_2=I$ but I cannot seem to find an example.
On the Hilbert space $\mathbb{R}^2$, what is a concrete example of a POVM that is not a PVM?
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quantum-information
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4Please consider spelling out acronyms. – 2017-01-13
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0Why not take F_1 to be the diagonal matrix with .5 and .4 on the diagonal and F_2 the diagonal matrix with .5 and .6? – 2017-01-13
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0There's a POVM that distinguishes |0> from |0>+|1> without making "silent" errors (it has a "distinguishing failed, input lost" result instead). Pretty sure the corresponding PVM requires ancilla and prep operations. – 2017-01-13
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0To expand Martin's comment, you need also to specify the borel sets that map to either one of the two matrices. The easiest example would be, as a measure on $\mathbb {R} $, the measure $\mu $ such that $\mu ([0])=F_1$ and $\mu ([1]) =F_2$ and $\mu (B) =0$ for any other borel set $B $ disjoint from those two. – 2017-01-14
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0Might [math.se] (maybe even [mathoverflow.se]?) be better suited for this question? – 2017-01-14
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0Acronyms: "Positive Operator Valued Measure" and "Projection Valued Measure". – 2017-01-17
1 Answers
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To do an example as you want, you take $F_1$ to be any positive matrix with $F_1\leq I$ (equivalently, $F_1$ is selfadjoint, and its eigenvalues are in $[0,1]$), and then take $F_2=I-F_1$. As a simplest example you could take $$ F_1=\begin{bmatrix}1/2&0\\0&1/3\end{bmatrix},\ \ \ F_2=\begin{bmatrix}1/2&0\\0&2/3\end{bmatrix}. $$